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Finsler geometry

A metric generalization of Riemannian geometry, where the general definition of the length of a vector is not necessarily given in the form of the square root of a quadratic form as in the Riemannian case. Such a generalization was first developed in the paper by P. Finsler [1].

The object studied in Finsler geometry is a real -dimensional differentiable manifold (of class at least ) with a system of local coordinates , on which a real non-negative scalar function in independent variables and is given, where are the components of the contravariant vectors tangent to at the point . Suppose that belongs to the class in , and that in each tangent space to there is a domain such that, first, it is conical (in the sense that if any vector tangent at some point belongs to , then every other tangent vector that is collinear with and tangent at the same point also belongs to ), and secondly, belongs to the class in . Non-zero vectors are called admissible. Suppose further that for every admissible and every point :

and also that is positively homogeneous of degree one in , that is, for every and all and admissible . Under these conditions the triple is called a Finsler space, and a Finsler metric. The value of is interpreted as the length of the vector tangent at .

If a Finsler space admits a coordinate system such that does not depend on these , then it is called a Minkowski space. The latter is related to a Finsler space in the same way as a Euclidean space is related to a Riemannian space. A Finsler space is called positive definite if one imposes a condition on that ensures that the quadratic form is positive definite for all and non-zero .

Imposing the condition of homogeneity in on has a clear geometrical meaning from the point of view of invariant concepts in centro-affine spaces, the tangent spaces being such spaces. Namely, the ratio of the lengths of any two collinear vectors and in can be invariantly defined in the following way: , which does not include any metric functions. Thus, the homogeneity condition imposed on is a condition that the Finslerian definition of length is consistent with the particular centro-affine definition; the Finsler metric is needed to compare the lengths of non-collinear vectors.

The tensor

is called the Finsler metric tensor. By Euler's theorem on homogeneous functions,

where, by definition, . These formulas are an immediate generalization of their Riemannian analogues, and follow from just the homogeneity condition. Finsler geometry reduces to Riemannian geometry in the case when the metric tensor is assumed to be independent of . The last condition can be written in the form , where

is called the Cartan torsion tensor. It satisfies the identity . All Finsler relations can be turned into their Riemannian analogues by setting . The Christoffel symbols , which are constructed from the Finsler metric tensor by the same formula as in Riemannian geometry, do not obey the transformation law of the coefficients of a connection. Nevertheless, one can construct the coefficients of a connection from the first derivatives of the Finsler metric tensor so that (as also in Riemannian geometry) the covariant derivative of the metric tensor vanishes. They are called the Cartan connection coefficients and have the form

where

From the commutators of various covariant derivatives one can find expressions for the Finsler curvature tensors.

In each tangent space the Finsler metric defines an -dimensional hypersurface (where the are regarded as fixed and the as varying), called the indicatrix. The indicatrix is formed by the ends of the unit tangent vectors tangent at the point . The fundamental significance of the concept of the indicatrix is already evident from the fact that, because the Finsler metric is homogeneous, the indicatrix at uniquely determines the form of at this point . In the Riemannian case the indicatrix is a sphere. Generally speaking, the indicatrix of a Finsler space can be a surface of a rather general form. The Finsler metric tensor induces a Riemannian metric on the indicatrix, converting it into a Riemannian space. For each fixed the Finsler metric tensor is Riemannian in the variables . The pair , where the are fixed and the are variable, is called the tangent Riemannian space at (a Euclidean space in the case of Riemannian geometry); the Riemannian curvature tensor of this space reduces to the expression . The indicatrix is a hypersurface that is imbedded in the tangent Riemannian space. The most immediate example of a Finsler metric function is the -th root of a form of order .

Let and be real scalar functions of class satisfying at each point the conditions or 2, and , and let be linearly independent real covariant vector fields of class , . Then for

the curvature of the indicatrix is constant and equal to , and for

the curvature tensor of the indicatrix is zero. The determinant of the Finsler metric tensor is independent of if and only if , where . If a Finsler space is positive definite and the indicatrix is a convex surface, then . The function is the only known example (1984) of a Finsler metric for which (not counting the proper Riemannian case).

One can select special types of Finsler spaces by postulating some special form of the characteristic Finsler tensors. If the base manifold admits a field of frames globally, and is the metric function of some Minkowski space, then one can introduce a Finsler metric on :

In this case the Finsler space and the metric are called -form. The functions and are -form when and are constants. -form spaces may be reckoned to be the simplest from the point of view of the way the variables enter in the metric. A Finsler space is called -reducible if it is not Riemannian, if and if the Cartan torsion tensor can be represented in the form

where . -reducible spaces can have metrics of only two types: either the Kropina metric , or the Randers metric , where , , is a covariant vector field, and is a Riemannian metric tensor. For example, the Lagrange function of an electric test charge in a gravitational or electromagnetic field is a Randers metric. The Finsler metric tensor corresponding to has signature , which makes it of interest in developing a Finslerian generalization of the general theory of relativity; this signature is also encountered in the case of the choice of metric tensors of the form . Such a generalization can be based on the concept of an oscillating Riemannian space to a Finsler space, according to which the Finsler metric tensor associates with each vector field the so-called osculating Riemannian metric tensor . Choosing tensor fields depending only on the , from which one constructs the Finsler metric according to , where is a scalar function, one can regard the as genuine gravitational field variables. The Finslerian geometrization of space-time also makes it possible to develop a theory of physical fields with various internal symmetries, relying on the concept of the group of transformations of the tangent vectors that leave the Finsler metric invariant.